Problem: Simplify and expand the following expression: $ \dfrac{4z}{5z - 7}+\dfrac{8}{z + 5} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5z - 7)(z + 5)$ Multiply the first term by $\dfrac{z + 5}{z + 5}$ $ \begin{align*} \dfrac{4z}{5z - 7} \times \dfrac{z + 5}{z + 5} & = \dfrac{(4z)(z + 5)}{(5z - 7)(z + 5)} \\ & = \dfrac{4z^2 + 20z}{(5z - 7)(z + 5)}\end{align*} $ Multiply the second term by $\dfrac{5z - 7}{5z - 7}$ $ \begin{align*} \dfrac{8}{z + 5} \times \dfrac{5z - 7}{5z - 7} & = \dfrac{(8)(5z - 7)}{(z + 5)(5z - 7)} \\ & = \dfrac{40z - 56}{(z + 5)(5z - 7)}\end{align*} $ Now we have: $ = \dfrac{4z^2 + 20z}{(5z - 7)(z + 5)} + \dfrac{40z - 56}{(z + 5)(5z - 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4z^2 + 20z + 40z - 56}{(5z - 7)(z + 5)} $ $ = \dfrac{4z^2 + 60z - 56}{(5z - 7)(z + 5)}$ Expand the denominator: $ = \dfrac{4z^2 + 60z - 56}{5z^2 + 18z - 35}$